The present invention relates to a new and distinct variety of peach tree named ‘NJ355’. Our new tree resulted from open pollination of the seed parent, J19-19-862144. The new variety differs from seed parent J19-19-862144 (unpatented) in that fruit of the new variety ripen approximately 3 weeks earlier and the flesh of the fruit clings to the pit when it is fully ripe. In comparison to the commercial peach variety ‘Sugar May’ (U.S. Plant Pat. No. 8,034), the fruit of the new variety has a lighter red blush, is lower in acidity, and matures approximately 6 days later. The fruit and leaves of the new variety are also more tolerant to bacterial leaf spot than ‘Sugar May’. The resulting tree was selected when growing in a cultivated area as the 44th tree in the 4th row of Block H at a fruit research center in Cream Ridge, N.J.